Calculus III: Taylor Series

Abstract

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n-1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K-theory.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper19.abs.html

Cited by in corpus (43)

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• The tangent complex and Hochschild cohomology of E_n-rings
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• Bar constructions for topological operads and the Goodwillie derivatives of the identity
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• Context-free manifold calculus and the Fulton-MacPherson Operad
• Calculus of functors and model categories
• Homotopical resolutions associated to deformable adjunctions
• Homotopy completion and topological Quillen homology of structured ring spectra
• Product and other fine structure in polynomial resolutions of mapping spaces
• Model Categories for Orthogonal Calculus
• On the Taylor Tower of Relative K-theory
• Goodwillie's Calculus of Functors and Higher Topos Theory
• Cross-effects and the classification of Taylor towers
• Homotopy nilpotent groups
• Homotopy theory of G-diagrams and equivariant excision
• The mod 2 homology of infinite loopspaces
• A streamlined proof of Goodwillie's n-excisive approximation
• Equivariant calculus of functors and Z/2-analyticity of real K-theory
• Central reflections and nilpotency in exact Mal'tsev categories
• Coassembly and the $K$-theory of finite groups
• Duality and small functors
• The Whitehead Conjecture, the Tower of S^1 Conjecture, and Hecke algebras of type A
• Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology
• Multiplicative Structure in the Stable Splitting of $\Omega S{L}_n(\mathbb{C})$
• Stability of concordance embeddings
• A multiplicative comparison of MacLane homology and topological Hochschild homology
• Homotopy theory of relative simplicial presheaves
• Infinity-operads and Day convolution in Goodwillie calculus
• A tower connecting gauge groups to string topology
• Polynomial functors in manifold calculus
• The Goodwillie tower for S^1 and Kuhn's theorem
• On automorphisms of high-dimensional solid tori
• The spectrum of excisive functors
• TQ-completion and the Taylor tower of the identity functor
• The Arone-Goodwillie spectral sequence for ${\Sigma }^{\infty }{\Omega }^n$ and topological realization at odd primes
• A classification of small linear functors
• A variant of a Dwyer-Kan theorem for model categories
• Homotopical Intersection Theory, III: multi-relative intersection problems
• Derived Character Maps of Groups Representations
• Cofibrantly generated model structures for functor calculus
• Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product
• Cosimplicial models for the limit of the Goodwillie tower